Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays

(1)

http://dx.doi.org/10.4236/jamp.2015.311175

Forced Oscillation of Nonlinear Impulsive

Hyperbolic Partial Differential Equation

with Several Delays

Vadivel Sadhasivam, Jayapal Kavitha, Thangaraj Raja

Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, India

Received 17 October 2015; accepted 24 November 2015; published 27 November 2015

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract

In this paper, we study oscillatory properties of solutions for the nonlinear impulsive hyperbolic equations with several delays. We establish sufficient conditions for oscillation of all solutions.

Keywords

Oscillation, Hyperbolic Equation, Impulsive, Delays

1. Introduction

The theory of partial functional differential equations can be applied to many fields, such as biology, population growth, engineering, control theory, physics and chemistry, see the monograph [1] for basic theory and applica-tions. The oscillation of partial functional differential equations has been studied by many authors see, for ex-ample [2]-[7], and the references cited therein.

The theory of impulsive partial differential systems makes its beginning with the paper [8] in 1991. In recent years, the investigation of oscillations of impulsive partial differential systems has attracted more and more at-tention in the literature see, for example [9]-[13]. Recently, the investigation on the oscillations of impulsive partial differential systems with delays can be found in [14]-[19].

(2)

( )

(

( )

)

( ) ( )

( )

(

( )

)

( )

(

)

( )

(

)

( )

(

)

1

, , , ,

, , , , , ,

, , , , ,

, , , , , , 1, 2, .

n

j j j k

j

k k k k

t k k k t k k

r t u x t a t u x t p x t f u x t

t t

q x t f u x t F x t t t x t G

u x t x t u x t

u x t x t u x t t t k

σ

α β

+

= +

+

∂  ∂ _{ =} _∆ ₋ 



 

∂  ∂  _

 

− − + ≠ ∈Ω× =

 

= 

 

= = = _

∑





(1)

with the boundary conditions

( )

( ) ( )

, , ,

u

h x u g x t x t

γ

+ ∂

+ = ∈ ∂Ω×

∂  (2)

( ) ( )

, , ,

u=ϕ x t x t ∈ ∂Ω×+ (3)

and the initial condition

( )

,

( ) ( )

, , ,

[

, 0

]

.

u x t = Φ x t x t ∈ −δ × Ω (4)

Here Ω ⊂N is a bounded domain with boundary ∂Ω smooth enough and ∆ is the Laplacian in the Euclidean N-space N, γ is a unit exterior normal vector of ∂Ω, δ =max

{ }

σj ,

( )

2

(

[

]

)

, , 0 , .

x t C δ

Φ ∈ − × Ω_

In the sequal, we assume that the following conditions are fulfilled:

(H1) r t

( ) ( )

,a t ∈PC

(

 +, +

)

, σ_j is a positive constant, p x t

( ) ( )

, , ,qj x t are class of functions which are piece wise continuous in t with discontinuities of first kind only at t=t_k,k=1, 2, and left continuous at

, 1, 2,

k

t=t k= 

(H2) f u

( ) ( )

, ,fj u C

(

)

+ +

∈ _{ } ; f u

( )

C

u ≥ is a positive constant,

( )

j

f u C

u ≥ is a positive constant, for

0;

u≠ h x

( )

∈C

(

∂Ω× Ω+,

)

; F x t

( )

, ∈PC

(

+× Ω,

)

; g t x

( )

, and ϕ

( )

t x, ∈PC

(

+× ∂Ω,

)

; 1 2

0 k , lim k .

k

t t t t

→∞ < < <_< <_ = ∞

(H3) u x t

( )

, and their derivatives u x t_t

( )

, are piecewise continuous in t with discontinuities of first kind only at t=t_k,k=1, 2,, and left continuous at t=t_k, u x t

(

, k

)

u x t

( )

,k ,

−

= u x tt

(

,k

)

u x tt

( )

, k , −

= k=1, 2,.

(H4) αk

(

x t u x t, ,k

(

,k

)

, , ,βk

(

x t u x tk t

(

, k

)

PC

(

, ,

)

+

∈  × Ω ×  k=1, 2,, and there exist positive

con-stants a a b b_k, , , *_k _k _k* and bk ≤ak* such that for k=1, 2,,

(

)

* k , ,k

k k

x t

a α ξ a

ξ

≤ ≤

(

)

* , ,

. k k

k k

x t

b β ξ b

ξ

≤ ≤

Let us construct the sequence

{ } { }

{ }

, j k k k

t = t  t_σ where tkσ_j = +tk σj and tk<tk+1, k=1, 2,.

By a solution of problem (1), (2) ((1),(3)) with initial condition (4), we mean that any functionu x t

( )

, for which the following conditions are valid:

1. If − ≤ ≤δ t 0, then u x t

( )

, = Φ

( )

x t, .

2. If 0≤ ≤ =t t1 t1, then u x t

( )

, coincides with the solution of the problem (1) and (2) ((3)) with initial con-dition.

(3)

4. If tk< ≤t tk+1, tk∈

{ }

tkσj , then u x t

( )

, coincides with the solution of the problem (2) ((3)) and the follow-ing equations

( )

(

( )

)

( )

(

( )

)

( )

(

)

( )

1

, , , ,

, , , , , ,

n

j j j k

j

t t

q x t f u x t σ F x t t t x t G

+ + +

+ ₊

=

∂  ∂ _{ =} _∆ ₋

 

∂  ∂ 

−

∑

− + ≠ ∈Ω×_ =

( )

,

(

,

)

,

( )

,

(

,

)

, for

{ }

\

{ }

, j

k k t k t k k k k

u x t+ =u x t u x t+ =u x t t ∈ t t_σ

or

( )

, , ,

(

,

)

,

( )

, , ,

(

,

)

, for

{ }

.

i i j

k k k k t k k k t k k k k

u x t+ =α x t u x t u x t+ =β x t u x t t ∈ t_σ  t

Here the number ki is determined by the equality tk=tki. We introduce the notations:

( )

(

)

{

, : , 1 ,

}

; 0 ,

k x t t t tk k x k k

∞

+ =

Γ = ∈ ∈Ω Γ =

_

Γ

( )

(

)

{

, : , 1 ,

}

; 0 ,

k x t t t tk k x k k

∞

+ =

Γ = ∈ ∈Ω Γ =

_

Γ

( )

min

( )

, and _j

( )

min _j

( )

, .

x x

p t p x t q t q x t

∈Ω ∈Ω

= =

The solution u∈C2

( )

Γ C1

( )

Γ of problem (1), (2) ((1),(3)) is called nonoscillatory in the domain G if it is either eventually positive or eventually negative. Otherwise, it is called oscillatory.

This paper is organized as follows: Section 2, deals with the oscillatory properties of solutions for the problem (1) and (2). In Section 3, we discuss the oscillatory properties of solutions for the problem (1) and (3). Section 4 presents some examples to illustrate the main results.

2. Oscillation Properties of the Problem (1) and (2)

To prove the main result, we need the following lemmas.

Lemma 2.1. Suppose that λ is the minimum positive eigenvalue of the problem

( )

x

( )

x 0, x ,

η λη

∆ + = ∈Ω

( ) ( )

0, ,

h x x x

η _η

γ

∂ ₊ ₌ _∈∂Ω

∂

and η

( )

x is the corresponding eigenfunction of λ. Then λ>0 and η

( )

x >0,x∈Ω. Proof. The proof of the lemma can be found in [20]. 

Lemma 2.2. Let _{u x t}

( )

_, _∈_C2

( )

_Γ __C1

( )

_Γ

be a positive solution of the problem (1), (2) in G. Then the func-tions

( )

( ) ( )

, d and

( ) ( )

, d 0

v t =

∫

_Ωu x t η x x

∫

_ΩF x t η x x≤

are satisfies the impulsive differential inequality

( ) ( )

( )

(

)

( )

1

, n

j j j k

j

r t v t λa t v t Cp t v t C q t v t σ R t t t =

′

′ + + + − ≤ ≠

 

 

∑

(5)

( )

* k

k k

k

v t

a a

v t +

≤ ≤ (6)

( )

*

, 1, 2,

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

(4)

where

( )

( ) ( )

, d .

R t =a t

∫

_∂Ωη x g x t S has an eventually positive solution.

Proof. Let u x t

( )

, be a positive solution of the problem (1), (2) in G. Without loss of generality, we may assume that there exists a T>0, t0>T such that u x t

( )

, >0, ,u x t

(

−σj

)

>0, 1, 2,j= , ,n for

( )

x t, ∈Ω×

[

t0,∞

)

.

For t≥t t0, ≠t kk, 1, 2,= , multiplying Equation (1) with η

( )

x , which is the same as that in Lemma 2.1

and then integrating (1) with respect to x over Ω yields

( )

( ) ( )

( )

( ) ( )

( )

(

( )

)

( )

(

)

( )

( ) ( )

1

d d

, d , d , , d

d d

, , d , d .

n

j j j

j

r t u x t x x a t u x t x x p x t f u x t x x

t t

q x t f u x t x x F x t x x

η η η

σ η η

Ω Ω Ω

= Ω Ω

 

= ∆ −

 

 

− − +

∫

∑∫

∫

By Green’s formula, and the boundary condition we have

( ) ( )

(

) (

)

(

)

( ) ( )

, d d d d d

, d , d ,

u

u x t x x u S u x g hu u h S u x

x g x t S u x t x x

η

η η η η η λη

γ γ

η λ η

Ω ∂Ω Ω ∂Ω Ω

∂Ω Ω

 ∂ ∂ 

∆ = _ − _ + ∆ = − − − + −

∂ ∂

 

= −

∫

where dS is the surface element on ∂Ω.

Also from condition (H2), and Jenson’s inequality we can easily obtain

( )

,

(

( )

,

)

( )

d

( ) ( ) ( )

, d

p x t f u x t η x x Cp t u x t η x x

Ω Ω

≥

∫

( )

,

(

,

)

( )

d

( )

(

,

)

( )

d

j j j j j j

q x t f u x t σ η x x C q t u x t σ η x x

Ω Ω

− ≥ −

∫

Thus, v t

( )

>0. Hence we obtain the following differential inequality

( )

( ) ( )

( )

( ) ( )

( ) ( ) ( )

( )

(

)

( )

( ) ( ) ( )

1

d d

, d , d , d

d d

, d , d ,

n

j j j

j

r t u x t x x a t u x t x x Cp t u x t x x

t t

C q t u x t x x a t g x t x S

η λ η η

σ η η

Ω Ω Ω

= Ω ∂Ω

 

+ +

 

 

− − ≤

∫

∑

∫

( ) ( )

( )

(

)

( )

1

, n

j j j k

j

′

′ + + + − ≤ ≠

 

 

∑

where

( )

( ) ( )

, d .

R t =a t

∫

_∂Ωη x g x t S For t≥t0, , 1, 2,t=tk k= , from (1) and condition (H4), we obtain

( )

(

)

* ,

, k

k k

k

u x t

a a

u x t +

≤ ≤

( )

(

)

* ,

. ,

t k

k k

t k

u x t

b b

u x t +

≤ ≤

According to v t

( )

u x t

( ) ( )

, η x d ,x

Ω

(5)

( )

* k

k k

k

v t

a a

v t +

≤ ≤

( )

*

, 1, 2, .

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

′ 

Hence, we obtain that v t

( )

>0 is a positive solution of impulsive differential inequalities (5)-(7). This completes the proof. 

Lemma 2.3. Let

( )

2

( )

1

( )

,

u x t ∈C Γ C Γ be a positive solution of the problem (1), (2) in G. If we further assume that f

( )

− = −u f u

( )

, u∈

(

0,+∞

)

and the impulsive differential inequality (5), and

( ) ( )

( )

(

)

( )

1

, n

j j j k

j

′

′ + + + − ≤ − ≠

 

 

∑

(8)

( )

* k

k k

k

v t

a a

v t +

≤ ≤ (9)

( )

*

, 1, 2,

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

′  (10)

have no eventually positive solution, then each nonzero solution of the problem (1)-(2) is oscillatory in the do-main G.

( )

, >0,u x t

(

, −σj

)

>0, 1, 2,j= ,n, for

( )

x t, ∈Ω×

[

t0,∞

)

.

From Lemma 2.2, it follows that the function v t

( )

is an eventually positive solution of the inequality (5) which is a contradictions.

If u x t

( )

, <0 for

( )

x t, ∈Ω×

[

t0,∞

)

, then the function

( )

,

( ) ( )

, , ,

[

0,

)

,

u x t = −u x t x t ∈Ω× t ∞

is a positive solution of the following impulsive hyperbolic equation

( )

(

( )

)

( ) ( )

( )

(

( )

)

( )

(

)

( )

1

, , , ,

, , , , , ,

n

j j j k

j

t t

q x t f u x t σ F x t t t x t + G

=

∂  ∂ _{ =} _∆ ₋

 

∂  ∂ 

−

∑

− + ≠ ∈Ω ×_ =

( )

,_k , ,_k

(

_k

(

, _k

)

,

u x t+ =α x t u x t

( )

,

(

, ,

(

,

)

, , 1, 2, . t k k k t k k

u x t+ =β x t u x t t=t k= 

( )

( ) ( )

, , , .

u

h x u g x t x t

γ

+

∂ ₊ _{= −} _∈∂Ω×

∂ 

and satisfies

( )

( ) ( )

( ) ( ) ( )

( )

(

)

( )

( ) ( ) ( )

1

d d

, d , d , d

d d

, d , d ,

n

j j j

j

t t

C q t u x t x x a t g x t x S

η λ η η

σ η η

Ω Ω Ω

= Ω ∂Ω

 

+ +

 

 

− − ≤ −

∫

∑

∫

  



( ) ( )

( )

(

)

( )

1

, n

j j j k

j

′

′ + + + − ≤ − ≠

 

(6)

where

( )

( ) ( )

, d .

R t =a t

∫

_∂Ωη x g x t S For t≥t t0, =t kk, =1, 2,, from (1) and condition (H4), we obtain

( )

(

)

* ,

, k

k k

k

u x t

a a

u x t +

≤  ≤



( )

(

)

* ,

. ,

t k

k k

t k

u x t

b b

u x t +

≤  ≤



According to v t

( )

u x t

( ) ( )

, η x d ,x

Ω =

_∫

  we obtain

( )

* k

k k

k

v t

a a

v t +

≤ ≤

( )

*

, 1, 2, .

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

′ 

Thus, it follows that the function v t

( )

u x t

( ) ( )

, η x dx

Ω =

_∫

  is a positive solution of the inequality (8)-(10) for

0

t >T which is also a contradiction. This completes the proof. 

Now, if we set g≡0 in the proof of Lemma 2.3, then we can obtain the following lemma.

Lemma 2.4. Let

( )

2

( )

1

( )

,

u x t ∈C Γ C Γ be a positive solution of the problem (1),(2) in G. If we further assume that f

( )

− = −u f u

( )

, u∈

(

0,+∞

)

( ) ( )

( )

(

)

1

0, n

j j j k

j

r t v t a t v t Cp t v t

C q t v t t t

λ

σ =

′

′ + +

 

 

+

∑

− ≤ ≠ (11)

( )

* k

k k

k

v t

a a

v t +

≤ ≤ (12)

( )

*

, 1, 2,

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

′  (13)

has no eventually positive solution, then each nonzero solution of the problem (1), satisfying the boundary con-dition

( )

0,

( )

, , k

u

h x u x t t t

γ

+

∂ ₊ ₌ _∈∂Ω× _≠

∂ 

is oscillatory in the domain G.

( )

, >0,u x t

(

, −σj

)

>0, 1, 2,j= , ,n for

( )

x t, ∈Ω×

[

t0,∞

)

.

From Lemma 2.2, it follows that the function v t

( )

is an eventually positive solution of the inequality (5) which is a contradiction.

(7)

( )

(

( )

)

( ) ( )

( )

(

( )

)

( )

(

)

( )

1

, , , ,

, , , , , ,

n

j j j k

j

t t

q x t f u x t σ f x t t t x t + G

=

∂  ∂ _{ =} _∆ ₋

 

∂  ∂ 

−

∑

− + ≠ ∈Ω ×_ =

( )

,k , ,k

(

k

(

, k

)

,

u x t+ =α x t u x t

( )

,

(

, ,

(

,

)

, , 1, 2, . t k k k t k k

u x t+ =β x t u x t t=t k= 

( )

0,

( )

, ,

u

h x u x t

γ

+

∂ ₊ ₌ _∈∂Ω×

∂ 

and satisfies

( )

( ) ( )

( ) ( ) ( )

( )

(

)

( )

1

d d

, d , d , d

d d

, d 0,

n

j j j

j

t t

C q t u x t x x

η λ η η

σ η

Ω Ω Ω

= Ω

 

+ +

 

 

− − ≤

∫

∑

∫

  



( ) ( )

( )

(

)

1

0, .

n

j j j k

j

r t v t λa t v t Cp t v t C q t v t σ t t =

′

′ + + + − ≤ ≠

 

 

∑

For t≥t t0, =t kk, =1, 2,, from (1) and condition (H4), we obtain

( )

(

)

* ,

, k

k k

k

u x t

a a

u x t +

≤  ≤



( )

(

)

* ,

, 1, 2, .

,

t k

k k

t k

u x t

b b k

u x t +

≤  ≤ = 



According to v t

( )

u x t

( ) ( )

, η x d ,x

Ω =

_∫

  we obtain

( )

*

* k

k k

k

v t

a a

v t

≤ ≤

( )

*

* k _.

k k

k

v t

b b

v t ′

≤ ≤

′

Thus it follows that the function v t

( )

u x t

( ) ( )

, η x d ,x

Ω =

_∫

  is a positive solution of the inequality (11)-(13) for

0

t >T which is also a contradiction. This completes the proof. 

Lemma 2.5. Assume that

(A1) the sequence

{ }

tk satisfies 0< < <t0 t1 , lim k k→∞t = ∞; (A2) m t

( )

∈PC1_ +, _ is left continuous at tk for k=1, 2,;

(A3) for k=1, 2, and t≥t0,

( )

( ) ( ) ( )

, _k,

m t′ ≤p t m t +q t t≠t

( )

_k _k

( )

_k _k,

m t+ ≤d m t +e

where p t

( ) ( )

, ,q t ∈C

(

 +

)

, d_k≥0 and ek are constants. PC denote the class of piecewise continuous

function from + to , with discontinuities of the first kind only at t=t k_k, 1, 2,= .

(8)

( )

(

( )

)

(

( )

)

( )

(

)

0 0

0

0 exp d exp d d

exp d .

k k

k k k j

t t t

k _t _t k _s

t t t s t t

t

j _t k

t t t t t t

m t m t d p s s d p r r q s s

d p s s e

< < < <

≤ +

+

∏

∫

∏

∫

∑ ∏

∫

Proof. The proof of the lemma can be found in [21]. 

Lemma 2.6. Let v t

( )

be an eventually positive (negative) solution of the differential inequality (11)-(13). Assume that there exists T≥t0 such that v t

( )

>0

(

v t

( )

<0

)

for t≥T. If

0 0

*

lim d

k

t k t t

t t s k

b s a

→+∞

∫

∏

_{< <} = +∞ (14) hold, then v t′

( )

≥0

(

v t′

( )

≤0

)

for t

[ ]

T t,1

(

k l

(

t tk, k1

]

)

,

+∞ + =

∈ _

_

where l=min

{

k t: _k≥T

}

.

Proof. The proof of the lemma can be found in [22].  We begin with the following theorem.

Theorem 2.1. If condition (14), and the following condition

( ) ( )

0 0

*

lim d ,

k

t _k

k t

t

t t s k

a

r t F s s b

→+∞

∫

∏

_{< <} = +∞ (15) hold, where

( )

_{( )}

(

( )

1

)

1

( )

exp

, n

j j

j

a t Cp t w t C q t

F t

r t

λ δ

=

+ + −

=

∑

then every solution of the problem (1), (2) oscillates in G.

( )

, be a nonoscillatory solution of (1), (2). Without loss of generality, we can assume that there exists T >0,t0≥T, such that u x t

( )

, >0, u x t

(

, −σj

)

>0, j=1, 2,,n for

( )

x t, ∈Ω×

[

t0,∞

)

.

From Lemma 2.4, we know that v t

( )

is a positive solution of (11)-(13). Thus from Lemma 2.6, we can find that v t′

( )

≥0 for t≥t0.

For t≥t0, t≠tk, k=1, 2,, define

( ) ( ) ( )

_{( )}

, 0.

v t

w t r t t t

v t ′

= ≥

Then we have w t

( )

>0, t≥t0, r t v t

( ) ( )

′ −w t v t

( ) ( )

=0. We may assume that v t

( )

0 =1, thus we have that for t≥t0

( )

(

( )

)

0

exp t d ,

t

v t =

∫

w s s (16)

( )

(

( )

)

0

exp t d ,

t

v t′ =w t

∫

w s s (17)

( )

(

( )

)

( )

(

( )

)

0 0

2

exp t d exp t d .

t t

v t′′ =w t

∫

w s s +w t′

∫

w s s (18)

Substitute (16)-(18) into (11) and then we obtain,

( ) ( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

0 0 0

2

1

exp d exp d exp d

exp d exp d exp j d 0.

t t t

n

t t t

j j

t t t

j

r t w t w s s r t w t w s s w t w s s

a t w s s Cp t w s s C q t σ w s s

λ −

=

 

′ + _ + ′ _

 

+ + + ≤

∫

∑

∫

(9)

( ) ( ) ( ) ( )

2

( )

(

( )

)

1

exp d 0, ,

j

n _t

j j _t k

j

r t w t r t w t a t Cp t C q t w s s t t

σ

λ ₋

= ′

+ + + +

∑

−

_∫

≤ ≠

or

( ) ( )

( )

(

( )

)

1

exp d 0, .

j

n _t

j j _t k

j

r t w t a t Cp t C q t w s s t t

σ

λ ₋

=

′ + + +

∑

−

_∫

≤ ≠

From above inequality and condition bk ≤a*k, it is easy to see that the function w t

( )

is nonincreasing for

1 0.

t≥ ≥ +t δ t Thus w t

( )

≤w t

( )

1 for t≥t1 which implies that

( ) ( )

( )

(

( )

1

)

( )

1

exp 0, .

n

j j k

j

r t w t λa t Cp t δw t C q t t t =

′ + + + −

∑

≤ ≠

From (12)-(13), we obtain

( ) ( ) ( )

_{( ) ( )}

*

_{( )}

( )

*

( )

,

k k k k

k k k k k

k k k

k

v t b v t b

w t r t r t r t w t

a v t a

v t

+

+ + +

+

′ ′

= ≤ =

and

( ) ( )

( )

(

( )

1

)

( )

1

exp , .

n

j j k

j

r t w t λa t Cp t δw t C q t t t =

′ ≤ − − − −

∑

≠

( )

*

( )

, 1, 2,

k

k k k

k

b

w t r t w t k a

+ _≤ ₌



Let

( )

_{( )}

(

( )

1

)

1

( )

exp

. n

j j

j

a t Cp t w t C q t

F t

r t

λ δ

=

− − − −

− =

∑

Then according to Lemma 2.5, we have

( )

( ) ( )

( )

0 0

0

0 0

0 * *

*

0 *

d

d 0.

k k

t

k k

k _t k

t t t k s t t k

t

k k

k _t k

t t t k t t s k

b b

w t w t r t r t F s s

a a

b a

w t r t r t F s s

b a

< < < <

≤ +

 

= _ − _<

 

∏

∫

∏

∫

∏

Since w t

( )

≥0, the last inequality contradicts condition (15). This completes the proof. 

3. Oscillation Properties of the Problem (1) and (3)

Next we consider the problem (1) and (3). To prove our main result we need the following lemmas.

Lemma 3.1. Suppose that λ0 is the smallest positive eigen value of the problem

( )

0

0, ,

x x x

x x

λ

∆Ψ + Ψ = ∈Ω

Ψ = ∈∂Ω

and Ψ

( )

x is the corresponding eigen function of λ0. Then λ0>0 and Ψ

( )

x >0, .x∈Ω

Proof. The proof of the lemma can be found in [20]. 

Lemma 3.2. Let

( )

2

( )

1

( )

,

u x t ∈C Γ C Γ be a positive solution of the problem (1), (3) in G. Then the func-tion

( ) ( )

, d 0

F x t x x

Ω Ψ ≤

∫

(10)

( ) ( )

( )

(

)

( )

0

1

, n

j j j k

j

r t v t a t v t Cp t v t

C q t v t Q t t t

λ

σ =

′

′ + +

 

 

+

∑

− ≤ ≠ (19)

( )

* k

k k

k

v t

a a

v t +

≤ ≤ (20)

( )

*

, 1, 2,

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

′  (21)

where

( )

, d .

Q t a t x t S

N

ϕ

∂Ω

∂Ψ = −

∂

∫

has the eventually positive solution

( )

( ) ( )

, d .

v t =

∫

_Ωu x t Ψ x x

( )

, be a positive solution of the problem (1), (3) in G. Without loss of generality, we may assume that there exists a T>0,t0>T such that u x t

( )

, >0,u x t

(

, −σj

)

>0, 1, 2,j= , ,n for

( )

x t, ∈Ω×

[

t0,∞

)

.

For t≥t t₀, , 1, 2,≠t k_k = , multiplying equation (1) with Ψ

( )

x , which is the same as that in Lemma 3.1 and then integrating (1) with respect to x over Ω yields

( )

( ) ( )

( )

( ) ( )

( )

(

( )

)

( )

(

)

( )

( ) ( )

1

d d

, d , d , , d

d d

, , d , d .

n

j j j

j

r t u x t x x a t u x t x x p x t f u x t x x

t t

q x t f u x t σ x x F x t x x

Ω Ω Ω

= Ω Ω

 

Ψ = ∆ Ψ − Ψ

 

 

− − Ψ + Ψ

∫

∑∫

∫

By Green’s formula, and the boundary condition we have

( ) ( )

( )

(

)

( )

(

( )

)

0

, d d d

, d d

, d , d ,

u

u x t x x u S u x

x t S u x

x t S u x t x x

γ γ

ϕ λ

γ

ϕ λ

γ

Ω ∂Ω Ω

∂Ω Ω

 ∂ ∂Ψ

∆ Ψ = _Ψ − _ + ∆Ψ

∂ ∂

 

∂Ψ

= − + − Ψ

∂ ∂Ψ

= − − Ψ

∂

∫

where dS is the surface element on ∂Ω. From condition (H2), we can easily obtain

( )

,

(

( )

,

)

( )

d

( ) ( ) ( )

, d

p x t f u x t x x Cp t u x t x x

Ω Ω

Ψ ≥ Ψ

∫

( )

,

(

,

)

( )

d

( )

(

,

)

( )

d .

j j j j j j

q x t f u x t σ x x C q t u x t σ x x

Ω Ω

− Ψ ≥ − Ψ

∫

The proof is similar to that of Lemma 2.1 and therefore the details are omitted. 

Lemma 3.3. Let u x t

( )

, ∈C2

( )

Γ C1

( )

Γ be a positive solution of the problem (1), (3) in G. If we further assume that f

( )

− = −u f u

( )

, u∈

(

0,+∞

)

( ) ( )

0

( ) ( )

( )

(

)

( )

1

, n

j j j k

j

r t v t λa t v t Cp t v t C q t v t σ Q t t t =

′

′ + + + − ≤ − ≠

 

(11)

( )

* k

k k

k

v t

a a

v t +

≤ ≤ (23)

( )

*

, 1, 2,

k

k k

k

v t

b b k

v t + ′

≤ ≤ =

′  (24)

have no eventually positive solution, then each nonzero solution of the problem (1), (3) is oscillatory in the do-main G.

Proof. The proof is similar to Lemma 2.3, and hence the details are omitted.  Futhermore, if we set ϕ≡0, then we have the following lemma.

Lemma 3.4. Let u x t

( )

, ∈C2

( )

Γ C1

( )

Γ be a positive solution of the problem (1), (3) in G. If we further assume that f

( )

− = −u f u

( )

, u∈

(

0,+∞

)

( ) ( )

0

( ) ( )

( )

(

)

1

0, n

j j j k

j

r t v t λ a t v t Cp t v t C q t v t σ t t =

′

′ + + + − ≤ ≠

 

 

∑

(25)

( )

* k

k k

k

v t

a a

v t +

≤ ≤ (26)

( )

* k _, _{1, 2,}

k k

k

v t

b b k

v t + ′

≤ ≤ =

′  (27)

has no eventually positive solution, then each nonzero solution of the problem (1), satisfying the boundary con-dition

( )

0, , , k

u= x t ∈∂Ω×+ t≠t

is oscillatory in the domain G.

Proof. The proof is similar to Lemma 2.4, and hence the details are omitted.  Using the above lemmas, we prove the following oscillation result.

Theorem 3.1. If condition (14) and the following condition

( ) ( )

0 0

*

< <

lim d = ,

k

t _k

k t

t

t t s k

a

r t F s s b

→+∞

∫

∏

+∞ (28)

hold, where

( )

0

( )

_{( )}

( )

,

a t Cp t F t

r t

λ +

=

then every solution of the problem (1), (3) oscillates in G.

( )

, be a nonoscillatory solution of (1), (3). Without loss of generality, we can assume that there exists T>0, ,t0≥T such that u x t

( )

, >0, u x t

(

, −σj

)

>0, j=1, 2,,n for

( )

x t, ∈Ω×

[

t0,∞

)

.

From Lemma 3.4, we know that v t

( )

is a positive solution of (25)-(27). Thus from Lemma 2.6, we can find that v t′

( )

≥0 for t≥t0.

For t≥t0, t≠tk, k=1, 2,, define

( ) ( ) ( )

_{( )}

, 0.

v t

w t r t t t

v t ′

= ≥

Then we have w t

( )

>0, t≥t₀, r t v t

( ) ( )

′ −w t v t

( ) ( )

=0. We may assume that v t

( )

0 =1, thus we have that for t≥t0

( )

(

( )

)

0

exp t d ,

t

(12)

( )

(

( )

)

0

exp t d ,

t

v t′ =w t

∫

w s s (30)

( )

(

( )

)

( )

(

( )

)

0

2

exp d

exp d .

t

v t w t w s s

w t w s s

′′ =

′ +

∫

(31)

We substitute (29)-(31) into (25) and can obtain the following inequality,

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

0 0

2

0

exp d exp d

exp d exp d 0,

t t

r t w t w s s w t w s s

a t w s s Cp t w s s

λ

 ₊ _′ 

 

 

+ + ≤

∫

then we have

( ) ( )

0

( )

0, k.

r t w t′ +λa t +Cp t ≤ t≠t

From (26)-(27), we can obtain

( ) ( ) ( )

_{( ) ( )}

_{( )}

( )

*

* , 1, 2, .

k k k

k

k k

k

v t b v t

w t r t r t

a v t v t

b

r t w t k

a

+

+ + +

+

′ ′

= ≤

= = 

It follows that

( ) ( )

0

( )

, k.

r t w t′ ≤ −λ a t −Cp t t≠t

( )

*

( )

* , .

k

k k k k

k

b

w t r t w t t t a

≤ =

Let

( )

0

( )

_{( )}

( )

.

a t Cp t F t

r t

λ

− −

− =

Then according to Lemma 2.5, we have

( )

( ) ( )

( )

0 0

0

0 0

0 * *

*

0 *

d

d 0.

k k

t

k k

k _t k

t t t k s t t k

t

k k

k _t k

t t t k t t s k

b b

w t w t r t r t F s s

a a

b a

w t r t r t F s s

b a

< < < <

≤ +

 

= _ − _<

 

∏

∫

∏

∫

∏

Since w t

( )

≥0, the last inequality contradicts (28). This completes the proof. 

Theorem 3.2. If condition (14) and the following condition

( )

₀ ₀

0 0

*

lim d ,

k

t k

k j j

t t

t t s k

a

r t C q s b

→+∞

∫

∏

_{< <} = +∞ (32) hold for some qj₀, then every solution of the problem (1), (3) oscillates in G.

Proof. The proof is obvious and hence the details are omitted. 

4. Examples

In this section, we present some examples to illustrate the main results.

(13)

(

)

(

( )

)

(

)

( ) ( ) (

)

( ) ( )

(

)

( )

(

) ( ) ( ) ( )

( )

(

)

( )

(

)

2 2 2 2 2

2

π , π sin , π cos ,

π

2 π sin , π sin 2 cos sin , 1, , 1, 2, 3,

2 1

, ,

, , , 1, 2,

k

k k

t k t k

t u x t t t u x t t t u x t

t t

t t u x t t t t x t t t k

k

u x t u x t k

u x t u x t k + +  ∂ ₊ ∂ _{= +} _∆ _{− +}    ∂  ∂  _   

− + _ − _− + > ≠ = 

  + =    = = _   (33)

and the boundary condition

( ) ( )

0, π, 0, 0, k, 1, 2, .

u t =u t = t≥ t≠t k=  (34)

Here

( )

* 1 *

( ) (

)

2

( ) (

)

2 2

( )

0,π ,a_k a_k k ,b_k b_k 1,k 1, 2, ,r t t π ,a t t π sin t ,

k +

Ω = = = = = = _ = + = +

( ) (

_{π cos}

)

2 2

( )

,

p t = +t t

( ) (

)

2

( )

1 1 1

π

2 π sin , , , ,

2

q t = t+ t σ = f u =u f u =u and taking

{ }

₁

{ }

1 2k . k

t +∞= +∞

Moreover

0 0

1 2 3 4

1 2 3

*

1 1

1

1 1 1 1

2 3

0

lim d d

1

d d d d

1 1 1 1

1 1 2 1 2 3 2

1 2 2 2

2 2 3 2 3 4 1

k k

k k k k

t k t t

t t s k t s

t t t t

t t t

t s t s t s t s

n

b k

s s

a k

k k k k

s s s s

k k k k

n

+ + +

+∞ →+∞ _{< <} _{< <}

< < < < < < < < +∞ = = + = + + + + + + + + = + × + × × + × × × + = = + ∞ +

∏

∫

∏

∫

∑

 

so (14) holds. We take ₀ 1, ₁ 1, max

{ }

₁ π,

( )

₁ 1

2 π

C C w t

t

λ = = = δ = σ = =

+ , then

( ) (

) (

)

( )

(

)

( )

2 2 2

2 2

π π cos

1 cos ,

π

t t t

F t t

t + + + = = + + thus

( ) ( )

(

)

(

( )

)

(

)

* 2 2 1 1 1 1 2 1 1

lim d lim 2 π 1 cos d

lim 1 cos d .

k k

t _k t _k

k

t t

t s k t s

t

a k

r t F s s s s

b k

s s →+∞ _{< <} →+∞ _{< <}

→+∞ + = + + ≥ + = + ∞

∏

∫

Hence (28) holds. Therefore all conditions of Theorem 3.1 are satisfied. Hence every solution of the problem (33), (34) oscillates in

( )

0,π ×

[

0,∞

)

. In fact u x t

( )

, =sin cosx t is one such solution of the problem (33) and (34).

Example 4.2. Consider the impulsive differential equation

(

)

(

( )

)

(

)

( ) ( ) (

)

( ) ( )

(

)

( )

(

)

( ) ( )

( )

(

)

( )

(

)

2 2 2 2 2

2 3

π , π sin , π cos ,

5π

2 π sin , 2 π cos cos , 1, , 1, 2, 3,

2

1

, ,

, , , 1, 2,

k

k k

t k t k

t u x t t t u x t t t u x t

t t

t t u x t t t x t t t k

k

u x t u x t k

u x t u x t k + +  ∂  ₊ ∂ _{= +} _∆ _{− +}    ∂  ∂  _   

− + _ − _+ + > ≠ = 

  + =    = = _   (35)

(14)

( )

0,

( )

π, 0, 0, , 1, 2, .

x x k

u t =u t = t≥ t≠t k=  (36)

Here Ω =

( )

0,π , * 1_,

k k

k a a

k +

= = *

1, k k

b =b = k=1, 2,.r t

( ) (

= +t π ,

)

2

( ) (

_{π sin}

)

2 2

( )

,

a t = +t t

( ) (

_{π cos}

)

2 2

( )

,

p t = +t t q t1

( ) (

=2 t+π sin

)

2

( )

t , 1 5π

, 2

σ = h u

( )

=0, f u

( )

=u f u, ,1

( )

=u and taking

{ }

1

{ }

2 1 . k k

t +∞= +∞ It is easy to check that the conditions of Theorem 2.1 are satisfied. Therefore, every solution of the problem (35), (36) oscillates in

( )

0,π ×

[

0,∞

)

. In fact u x t

( )

, =sin cost x is one such solution of the problem (35) and (36).

Acknowledgements

The authors thank Prof. E. Thandapani for his support to complete the paper. Also the authors express their sin-cere thanks to the referee for valuable suggestions.

References

[1] Wu, J.H. (1996) Theory of Partial Functional Differential Equations and Applications. New York, Springer. http://dx.doi.org/10.1007/978-1-4612-4050-1

[2] Liu, A.P. (1996) Oscillations of Certain Hyperbolic Delay Differential Equations with Damping Term. Mathematica Applicate,9, 321-324.

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